Conjecture that perfect groups $P(r,n,k,r-1,q)$ are of type $\tilde{\mathfrak{Z}}$

Let $P(r,n,k,s,q)$ be the cyclically presented group in question, and let $\tilde{\mathfrak{Z}}$ denote the group type used in the paper. Let $n,k,q\geq 1$ and $2\leq r<n$ be integers such that

$$(n,k-1,q)=1.$$

Assume also that

$$k\not\equiv 1\pmod n,\qquad k\not\equiv 1+q\pmod n.$$

Type-$\tilde{\mathfrak{Z}}$ conjecture. If $P(r,n,k,r-1,q)$ is perfect, then it is of type $\tilde{\mathfrak{Z}}$. The preceding theorem establishes the corresponding perfectness criterion within the type-$\tilde{\mathfrak{Z}}$ family; the conjecture proposes that this type condition is necessary for perfect, nontrivial groups under the stated hypotheses. The supplied text gives no resolution.